Definition 1. A family of Enriques surfaces is an algebraic space $f \colon Y \to S$ which is flat, proper, of finite presentation, such that each geometric fiber is an Enriques surface.
Proposition 2. Let $Y \to S = \operatorname{Spec}R$ be a family of Enriques surfaces. Then $\operatorname{Pic}_ {Y/S}$ is a scheme, its torsion part $\operatorname{Pic}_ {Y/S}^\tau$ is a flat group scheme of order $2$, and its numeric classes $\operatorname{Num}_ {Y/S} = \operatorname{Pic}_ {Y/S} / \operatorname{Pic}_ {Y/S}^\tau$ is representable by a local system of free abelian groups of rank $10$.
One observation is that the only flat group scheme over $\mathbb{Z}$ with order $2$ are $\mathbb{Z}/2\mathbb{Z}$ and $\mu_2$.
A general fact is that there is a canonical class in $H^0(S, Rf_\ast (G_Y))$ where $G = \operatorname{Hom}(\operatorname{Pic}_ {Y/S}^T, \mathbb{G}_ {m,S})$. Now if this class vanishes in $H^2(S, G)$ in the Leray spectral sequence exact sequence, then we will be able to lift it to $H^1(Y, G_Y)$ and we call this torsor $X \to Y$ a family of canonical coverings.
Proposition 3. A family of canonical coverings $\colon X \to Y$ exists if one of the following holds:
- $H^2(S, G) \to H^2(Y, G_Y)$ is injective,
- $Y \to S$ admits a section,
- $\operatorname{Pic}_ {Y/S}^\tau \to S$ is étale and $\operatorname{Pic}(S) = 0$.
The first two follow from the long exact sequence. For the third, we can check that $\mathscr{L} = \Omega^2_{Y/\operatorname{Spec} R}$ has trivial second power, and so we can just set $X = \operatorname{Spec}(\mathscr{O} \oplus \mathscr{L})$.
Proposition 4. If $S = \operatorname{Spec} \mathbb{Z}$ then $Y$ has constant Picard scheme.
Theorem 5. If $Y \to S = \operatorname{Spec} R$ has constant Picard scheme and $\operatorname{Pic}(S)$ and $\operatorname{Br}(S)$ vanish, then there exists a canonical covering $X \to Y$ and the set of isomorphisms classes of these form a principal homogenoeus space for $R^\times / (R^\times)^2$. For a point $\operatorname{Spec} k \to S$,
- $\omega_{Y_k}$ has order $2$,
- all classes $\ell \in \operatorname{Num}_ {Y_k/k}(\bar{k})$ come from $Y_k$, unique up to twisting by $\omega_{Y_k}$,
- every (-2)-curve $\bar{E} \subseteq Y_\bar{k}$ is the base change of a (-2)-curve $E \subseteq Y_k$ and $E \cong \mathbb{P}_ k^1$,
- every genus one fibration $Y_\bar{k} \to \mathbb{P}_ \bar{k}^1$ comes from $Y_k \to \mathbb{P}_ k^1$, which has exactly two multiple fibers (i.e., where fiber classes is some multiple of the reduced class), lying over $k$-rational points,
- there actually exists a genus one fibration $Y_k \to \mathbb{P}_ k^1$.